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107) by breaking the real line into two intervals or by using the generalized derivative F 0 [] = " Z d [h (x) lnjxj ](x) dx dx " (3:109) Here we are integrating oversupp and we do not worry about the terms coming from the limits of the integral in the integration by parts because = 0 at the limit points. 109): d h (x) [h (x)lnj xj] = " 0 ln"(x + ") + ln(")(x 0 ") dx " x 25 (3:110) Here and below, we have used the result that (x) (x 0 x0 ) = (x0 ) (x 0 x0). 109) and integrating with respect to x, we have F"0 [] = 0ln "(0 ") + ln(")(") + = (0)ln + Z h (x) " (x) dx Z h (x) " (x) dx + o(") x (3 :111) x where o(") stands for terms of order " and higher.

I = d dt Z (t ) ( ) (4:1) Q x; t d x where (t ) is a time-dependent region of space and Q(x; t) is a C1 function. Let us assume the boundary @ (t ) of is piecewise smooth and is given by the surface f = 0 such that f > 0 in . Assume also that rf = n 0 where n 0 is the unit inward normal to the surface. 1) is continuous in time. Then, we can replace =dt and bring the derivative inside the integral. We write d=d t with d Z d h(f )Q(x; t ) d x I = dt = = Z @f @t Z () ( f Q x; t @f @ (t) @t ( ) @Q ) + h(f ) @ t Q x; t dS + Z (t ) dx @Q @t dx (4:2) where h(f ) is the Heaviside function.

133, no. 1, 1989, p. 189. Response by F. Farassat and M. K. Myers (same journal and page). 41. Ffowcs Williams, J. ; and Hawkings, D. : Sound Generation by Turbulence and Surfaces in Arbitrary Motion. Philos. Trans. R. Soc. London, ser. A, vol. 264, no. 1151, May 8, 1969, pp. 321{342. 42. : Can Shock Waves on Helicopter Rotors Generate Noise? A Study of the Quadrupole Source. , vol. 1, 1990, pp. 323{346. 43. ; and Lee, Yung-Jang: Development of a Shock Noise Prediction Code for High-Speed Helicopters|The Subsonically Moving Shock.